Erschienen in: Studies in Nonlinear Dynamics & Econometrics ; 19 (2015), 4. - S

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1 Erschenen n: Studes n Nonlnear Dynamcs & Econometrcs ; 9 (5), 4. - S Stud. Nonlnear Dyn. E. 5; 9(4): 5 59 Ldan Grossmass* and Ser-Huang Poon Estmatng dynamc copula dependence usng ntraday data Abstract: We estmate the dynamc daly dependence between assets by applyng the Semparametrc Copula-Based Multvarate Dynamc (SCOMDY) model on ntraday data. Usng tck data of three stock returns of the perod before and durng the credt crss, we fnd that our dependence estmator better captures the steep ncrease n dependence durng the onset of the crss as compared to other commonly used tme-varyng copula methods. Lke other hgh-frequency estmators, we fnd that the dependence estmator exhbts long memory and forecast t usng a HAR model. We show that for out-of-sample forecasts, our dependence estmator performs better than the constant estmator and other commonly used tme-varyng copula dependence estmators. Keywords: copula; hgh frequency data; ntraday dependence; tme-varyng dependence; value-at-rsk. JEL classfcaton: C4; C8; C58; G7. DOI.55/snde-3-3 Introducton Ths paper harnesses the rch nformaton set n ntraday data to obtan a useful estmate for the daly tmevary ng copula dependence. We wsh to obtan from ntraday data the daly nonparametrc condtonal dependence that s measured ex-post and term ths the ntraday (copula) dependence. Our ntraday dependence measure s estmated usng the Semparametrc Copula-Based Multvarate Dynamc (SCOMDY) model (Chen and Fan 6) appled to ntraday data. In SCOMDY models, the ndvdual seres condtonal mean and condtonal varance s parametrcally specfed whle ther jont dstrbuton s specfed by a sem-parametrc copula (parametrc copula wth emprcal margnals). Chen and Fan (6) noted that n such models, the lmtng dstrbuton of the copula dependence parameters are unaffected by the frst-step estmaton of the dynamc parameters specfyng the condtonal mean and varance. It s however affected by the margnal dstrbuton of the standardzed resduals. Here, we adopt Chen and Fan (6) SCOMDY approach but apply t to ntraday data. We show emprcally that the derved dependence from ntraday data s a superor estmator of daly dependence as compared to commonly used tme-varyng copulas. Multvarate volatlty models have been a long-standng challenge n fnancal econometrcs, especally wth the estmaton and modelng of correlatons between tme seres. Recent fnancal crses have taught us that models based on normal dstrbuton are nadequate, and correlaton changes through tme and ncreases sharply durng crses. Ths paper brngs together two man areas of econometrc lterature, namely that of tme-varyng copulas and of econometrc estmators that harness hgh frequency data. Sklar (959) ntroduced a jont dstrbuton functon to be flexbly decomposed nto ts ndvdual margnals and a dependence functon, termed the copula functon, lnkng the margnals. By specfyng dfferent *Correspondng author: Ldan Grossmass, Department of Economcs, Unversty of Konstanz, Box 4, Konstanz, Germany, e-mal: rachelldan@gmal.com Ser-Huang Poon: Manchester Busness School, Crawford House, Unversty of Manchester, Manchester, UK Konstanzer Onlne-Publkatons-System (KOPS) URL:

2 5 L. Grossmass and S.-H. Poon: Estmatng dynamc copula dependence usng ntraday data copula functons, dfferent dependence structures beyond lnear correlatons can be formed and estmated. Ths makes copulas popular tools n fnance when dealng wth multvarate dstrbutons. A man shortcomng s that they are statc,.e., they assume..d data, and hence only consder the spatal dependence. In realty, fnancal tme seres often exhbt tme-varyng dependences and structural breaks [see, e.g., Patton ()]. It s thus mportant to use tme-varyng copula models. One of the most tradtonal methods of obtanng tme-varyng dependence s by usng rollng wndows of observatons for estmaton. However, ths method s arbtrary on the choce of wndow sze, s slow to react to market changes, and stll suffers from the effects of structural breaks. Developments n ths area nclude the regme-swtchng models of Rodrguez (7) and Chollete, Henen, and Valdesogo (9), and the autoregressve dependence parameter of Patton (6). The autoregressve dependence of Patton (6) assumes that the dependence parameter follows an ARMA(,) process, and the determnaton of the forcng (or updatng) varable and the choce of number of lags are key ssues to deal wth. To avod specfyng the parameter changng scheme, Gacomn, Hardle, and Spokony (9) use local change pont analyss to detect ntervals of local homogenety for estmaton of the copula parameter. In a more recent paper, Creal, Koopman, and Lucas (3) GAS model addresses the problem of choosng a forcng varable n Patton (6) by usng the lagged score whch then ncorporates nformaton of the entre densty. All these models face varyng degrees of the same problem: when too much data from the dstant past s used, the estmated parameters could be based. However usng only recent data would result n too few observatons for nference. Whle the local change pont analyss method attempts to crcumvent ths problem, t s only effectve f suffcent post-break data s avalable. Second, the avalablty of hgh-qualty tck-by-tck data n fnancal tme seres has led to much nnovaton and research n the last decade of fnancal econometrcs, wth the development of realzed volatlty of Barndorff-Nelsen and Shephard (), realzed covarances and other realzed measures based on the theory of quadratc varaton. The realzed correlaton [Barndorff-Nelsen and Shephard (4), henceforth BNS] s derved from dvdng realzed covarances by the realzed volatltes of the assets. We also wsh to harness ntraday data n estmatng correlatons and other non-lnear dependence measures for use n copulas for two smple reasons: () they provde nformaton on the most current status about the relatonshp between dfferent assets of nterest; () they provde a large quantty of observatons for econometrc nference. The dea of usng ntraday data for copula dependence estmaton has been consdered before, but was focused on samplng frequency that s shorter than day for use n estmatng dependence across a large number of days (.e., not wthn the same day). For example, Das and Embrechts (4) do not model the tme-varyng dependence parameter explctly, they apply the method of change-pont analyss on the copula dependence parameter. Breymann, Das, and Embrechts (3) consder FX data and fnd that the emprcal margnals of the resduals can be rejected for ellptcty for frequences of 8 h and shorter. In a dfferent and more recent approach, Fengler and Okhrn () makes use of Hoeffdng s lemma to match the realzed covarance to a copula dependence parameter usng methods-of-moments and term ths the realzed copula. Salvaterra and Patton (3) extend the GAS model of Creal, Koopman, and Lucas (3) by addng an addtonal regressor of realzed measures n the autoregresson equaton. Here we use daly returns n the SCOMDY model but wth daly tme-varyng dependence parameter estmated usng the same SCOMDY model on the ntraday data wthn the day. Our smulaton studes ndcate that the dependence parameter s stable under tme aggregaton and s based for non-gaussan copulas. However despte the bas, we show that such a method gves us a useful estmate of the daly dependence. We apply the method durng the crss perod between 6 and 8 where our dependence estmator pcks up the crss clearly and dramatcally as compared to other methods. An emprcal horse race also shows clear evdence that the ntraday dependence estmator beats exstng tme-varyng copula methods by provdng more accurate out-of-sample value-at-rsk (VaR) forecasts. The rest of ths paper s organzed as follows: Secton dscusses the class of semparametrc copulabased multvarate dynamc models, ntroduces the concept of ntraday dependence and dscusses ts propertes. Secton 3 descrbes two smulaton experments that renforces the dscussons n Secton. Secton 4

3 L. Grossmass and S.-H. Poon: Estmatng dynamc copula dependence usng ntraday data 53 begns the emprcal study, where we estmate the ntraday dependence measures for three stock pars and dscuss some of ts estmaton ssues. The dependence measures are then forecasted for the -step ahead VaR estmaton. Secton 5 concludes. Hgh frequency data and semparametrc copula Assume the returns processes y a and y b of assets a and b are..d. Let F ab be the jont densty functon between the returns. Sklar s theorem (Sklar 959) states that F ab can be decomposed usng nto the ndvdual margnal denstes and a copula, F ( y, y ) = C( F ( y ), F ( y ); θ), ab a b a a b b where F a and F b are contnuous margnals and C s a copula wth dependence parameter θ. The copula s thus a dependence functon between the two asset returns, and varous multvarate denstes can be descrbed by specfyng the copula functon. For rsk management purposes, an mportant copula concept s tal dependence, v+ C( v, v) where the upper and lower tal dependence are gven by τ = lm Pr( F ( y ) > vf ( y ) > v) = lm U v a a b b v v C( v, v) and τ = lm Pr( F ( y ) vf ( y ) v) = lm respectvely. These descrbe the dependence n the L v a a b b v v upper rght quadrant and lower left quadrant respectvely, or the probabltes that extreme events occur smultaneously. Gaussan copulas have no tal dependence and are often nadequate for modelng fnancal returns, where the lower tal dependence s typcally observed to be larger than upper tal dependences. The clayton and survval gumbel copulas are more sutable n ths respect. Fnancal returns are however not..d and typcally exhbt nter-temporal dependence. The SCOMDY model allows for the dynamcs (condtonal mean and condtonal varance) of the ndvdual return seres to be modeled, and the dependence of the scale-free resduals to be then specfed by a copula. Furthermore, rather than pre-specfyng the parametrc dstrbuton of the margnals, the emprcal margnal dstrbuton functons are used. Ths reduces the rsk of msspecfyng the margnals n the estmaton of dependence but comes at a cost of decreasng effcency of the estmaton (as compared to when the parametrc forms of the margnals are known). Chen and Fan (6) showed that n ths two-step procedure, the asymptotc dstrbuton of ˆθ s unaffected by the ntal step of GARCH estmaton. In ths paper, we estmate the tme-varyng daly dependence parameter from ntraday data wthn the day by applyng the same SCOMDY model on all ntraday returns n day : F ( η, η ) = C( F ( η ), F ( η ); θ ) () m m ab a,, j b,, j a a,, j b b,, j m where θ s the ntraday dependence for day, η a,,j and η b,,j are the ntraday resduals from a parametrc model for day and ntraday perod j. The superscrpt m denotes the samplng frequency at whch the estmaton takes place. The margnals F a and F b can be estmated by the emprcal CDF F n,a and F n,b respectvely. We post that explotng ntraday data allows us to observe the dynamcs of the latent condtonal dependence. Ths approach s much less senstve to structural breaks n long tme seres as only nformaton wthn the day s used. There are however some mportant ssues to contend wth. Except for the Gaussan copula, copula For detaled dscusson on copulas, refer to Nelsen (999) and Joe (997). A bref descrpton of the form of some commonly used copulas s gven n the Web Appendx, Secton E. See Genest, Ghoud, and Rvest (995) whch derves the asymptotc propertes of the canoncal maxmum lkelhood (CML) estmator. The term canoncal s used here as t s no longer a standard maxmum lkelhood estmaton due to the use of emprcal margnals.

4 54 L. Grossmass and S.-H. Poon: Estmatng dynamc copula dependence usng ntraday data dstrbutons such as Gumbel and Clayton are not closed under aggregaton,.e., the dstrbuton of returns at daly level and at ntradaly level are not the same. However, we show that by passng returns through a GARCH flter n the SCOMDY framework, the scale-free resduals retan ther dependence structure wth relatve stablty. It s thus mperatve for our purpose to consder the propertes of GARCH resduals. Another ssue s that hgh-frequency returns do not capture nformaton durng non-tradng hours, unlke the case when daly opento-open or close-to-close returns are used. Due to these ssues whch we wll dscuss n the followng subsecton, there s basness n the ntraday dependence estmator. We fnd however that t remans a useful method of estmatng daly dependence, and consder usng an adjustment factor for the estmator. Intraday seasonalty and other practcal estmaton ssues also have to be addressed and we do ths n Secton The ntraday dependence estmator Assume a unvarate ARMA(,)-GARCH(p,q) process s used to model the condtonal mean and varance of ntraday returns y k,,j, k =, K, =,, t, j =,, /m, for m regular tme-spaced samplng ntervals wthn a m m day, < m. Let us denote η,, η, t as the vector of GARCH resduals of K assets n day, where t, K, s the number of days n the sample perod. For each day and samplng frequency m, the ntraday return at tme j s gven by y c a y b () m m m = + + +, k,, j k, k, k,, j k,, j k, k,, j (3) =σ η, k,, j k,, j k,, j q p m m m = + + y k,, j k, k,, q k,, j q k,, p k,, j p q = p = σ ω β σ α, (4) where k =,, K, =,, t, j =,, /m and ω >, β >, α >. The parameters m, m, m, m c a b ω, k, k, k, k, m m m k, k,, q k,, p β and α for q =,, q, p =,, p are dfferent for each asset k, each day and are a functon of m, m m k,, q k,, p hence the correspondng subscrpts and superscrpt. The copula functon of the ntraday resduals s gven by F ( η,, η ) = C ( F ( η ),, F ( η ); θ ), (5) m m K,, j K,, j,,, j K, K,, j m where θ s the ntraday dependence on day for the K assets. The margnal dstrbutons F k, (η k,,j ) can be estmated by the emprcal margnal dstrbutons of the resduals for each day 3 ˆ m F ( x) = I( ˆη x), < x<, < m. (6) + m k, k,, j j / m+ where ˆη are the estmated standardzed resduals from (3). We now consder the constant dependence estmator, θ, for the perod. The constant dependence between K assets s gven by θ under a SCOMDY model, F ( η,, η ) = C( F( η ),, F ( η ); θ), K, K,, K K, where η,,,η K, are the vector of resduals from an ARMA(,)-GARCH(p,q) process at a daly samplng frequency: The margnal dstrbuton functon F k (x) can be estmated n a smlar fashon as (6). 3 Note that the counter for j begns from here as n Berkes and Horvath (3) as the frst estmated resdual tends to be based due to the ntal values assumed for the condtonal mean and varance. Other papers however nclude the frst estmated resdual,.e., j begns from.

5 L. Grossmass and S.-H. Poon: Estmatng dynamc copula dependence usng ntraday data 55 The copula CML estmators ˆθ and ˆm θ are gven by t ˆ θ= argmax log c( Fˆ( ˆη ),, Fˆ ( ˆη ); θ), θ, K K, t = and ˆ θ m arg log ( ˆ ( η ),, ˆ ( η ); θ ) / m+ m m = max ˆ ˆ c F F,,, j K, K,, j θ m + m j= respectvely, where c(x,, x k ; θ) s the copula densty functon. We have now defned our ntraday dependence estmator ˆ m θ. Consder the returns of two assets k and k and assume that dependence s non tme-varyng between the days = to = t and s gven by θ. What we are nterested n s how ˆm θ relates to ˆ. θ We consder two cases: () gaussan copulas and () non-gaussan copulas. We fnd that under certan crcumstances n the second case, ˆm θ would be based. () Gaussan case: For smplcty, assume that the returns follow a GARCH(,) process. 4 GARCH(,) models are compatble wth contnuous tme processes and can be vewed as a dscretsed form of contnuous stochastc models. 5 Berkes and Horvath (3) proved that the asymptotc behavor of the emprcal process of squared GARCH resduals weakly converges to a Gaussan process. Ths result was used n Chan et al. (9) to establsh the CML consstency usng weghted approxmaton of the emprcal resduals and to show that the lmtng dstrbuton of ˆ θ s ndependent of the GARCH flterng. Consder two assets k and k, let v = Φ ( F( η )) and v = Φ ( F( η )), where Φ denotes the standard normal cumulatve dstrbuton. In the Gaussan copula, the copula dependence parameter s smply the, k,, k, lnear correlaton between v, and v,. Thus t t v v v v =,, t t =, =, ˆθ = t v v t v v (7) where v t t v k = k, =. Usng ntraday data, let Gaussan copula dependence s gven by v = Φ ( F( η )) and,, j k,, j v = Φ ( F( η )), then the,, j k,, j ˆθ + v v v v + / m j,, j,, j,, m m = = + / m + / m + v v v v j=,, j, + j=,, j, m m (8) where v + / = + m v. m Usng Theorem C. (n the Web Appendx), derved under assumptons k, j= k,, j about the dstrbuton of GARCH resduals and usng the results of the emprcal process of GARCH resduals 4 The smple GARCH model for estmatng the condtonal varance s commonly used to model daly returns, see e.g., Km, Malz, and Mna (999), Chrstoffersen and Debold (6) and Gacomn, Hardle, and Spokony (9). These papers ponted out that for daly equty data, condtonal means are domnated by condtonal varances, hence a zero-mean assumpton s reasonable. We also checked our results by ncludng a condtonal mean ARMA(,) model. The condtonal means are small and overall results reman unchanged. Condtonal means matter more for ntraday data than for daly data, but are stll domnated by condtonal varances, as noted by Andersen and Bollerslev (997). We consder the use of ARMA(,) to model condtonal means but fnd that the assumpton of a zero condtonal mean s adequate. We dscuss ths ssue further n Secton Nelson (99) showed that a sequence of dscrete tme GARCH(,) processes wth..d. normal nnovatons converges n dstrbuton to Itô processes as the length of the dscrete tme ntervals goes to zero.

6 56 L. Grossmass and S.-H. Poon: Estmatng dynamc copula dependence usng ntraday data n Berkes and Horvath (3), lm ˆ / m sup x< F F ( x) = asymptotcally. It s then straghtforward to k, k see that both ˆm θ and ˆθ converges n probablty to θ as /m or t. () Non-Gaussan case: Theorem C. s based on the assumpton that the GARCH resduals at both frequences have a dstrbuton wth zero mean and unt varance, whch mples a strong GARCH process [see Drost and Njman (993)]. Ths assumpton s often made by researchers wthout regard to samplng frequency used. However, strong GARCH processes are not closed under aggregaton and only weak GARCH processes are tme-aggregatng (for weak GARCH, the estmated σ s not the condtonal varance but the best lnear predctor of the squared t resduals,.e., E( y ) r σ y =,, r =,,.). Ths means that f the data generatng process s weak t+ t t GARCH(,) at a certan samplng frequency, then the weak GARCH(,) wll be the data generatng process for any other samplng frequency (Alexander and Lazar 5). In a semnal paper, Drost and Njman (993) defned the strong, sem-strong and weak GARCH, where strong GARCH assumes the resduals to be..d. and have a dstrbuton wth zero mean and unt varance [e.g., N(, )]. Sem-strong GARCH assumes that the resduals are uncorrelated whle for weak GARCH, the estmated σ s not the condtonal varance but the best lnear predctor of the squared resduals,.e., t E( y ) r σ y =,, r =,,. Denote the parameter vector of the true GARCH process by γ t+ t t k, = (ω k, β k, α k ) T, and the parameter vector of the ntradaly GARCH process by γ m = ( ω m, β m, α m ) T. Wth these defntons, k k k k Drost and Njman (993) showed that GARCH models n the strong or sem-strong sense are not closed under m aggregaton and hence γ γ. Only weak GARCH models are tme-aggregatng. Ths means that f the data k k, generatng process s weak GARCH(,) at a certan samplng frequency, then the weak GARCH(,) wll be the data generatng process for any other samplng frequency (Alexander and Lazar 5). Nelson (99) showed that as samplng frequency ncreases, the GARCH resduals approach a lmtng Student s-t dstrbuton wth +4τ/α * degrees of freedom, where τ = lm Δt Δt - ( β * α * ). Alexander and Lazar (5) derve the contnuous lmt of the weak GARCH(,) as a stochastc volatlty model wth prce-volatlty correlaton that s related to the skewness and kurtoss of the returns densty. The lmt reduces to the results of Nelson (99) f the returns densty s normal. Smlarly, Drost and Werker (996) 6 show that the mplct assumpton of an underlyng contnuous model mples the presence of heavy tals, and consstently fnd that leptokurtoss s less pronounced n aggregated seres. Ths means that for any copula model whch captures tal dependences, ˆm θ wll be larger than ˆθ as samplng frequency ncreases. Despte the basness n the dependence estmate, the ntraday dependence estmator remans a good estmator for tme-varyng dependence for several reasons. Frst, smulatons n Drost and Njman (99) suggest that the QML estmates are close to the true parameters even f the model s weak GARCH. Second, Nelson (99) and Alexander and Lazar (5) showed that as samplng frequency ncreases, the estmated GARCH varance wll offer a good approxmaton for the varance of the true process, even f the GARCH model s msspecfed. Both of these arguments mply that the effects of tme aggregaton of GARCH s lmted. / m+ Fnally, n realty the sum of ntraday returns s often not equvalent to daly returns,.e., y. k, y j= k,, j Ths s because ntraday returns do not take nto account of overnght returns when tradng s not 4 hours, for example, n stock markets. Whle there are no ntraday observatons n non-tradng hours, stock prces do shft due to nternatonal news releases and other nformaton made avalable to the nvestor. Ths means that dependence between assets n non-tradng hours s not taken nto account by the ntraday dependence estmator and t wll be downward-based relatve to ˆ. θ There s also Epps effect to deal wth at hgh samplng frequences, where dependence s based to zero as samplng ntervals approaches zero. Ths phenomenon s due to non-synchronous tradng of assets, but can usually be dealt wth by usng lower frequency returns (e.g., or 5 mn returns) rather than the hghest samplng frequency. However, by usng lower frequency returns, the number of observatons wthn the sngle 6 Drost and Werker (996) also showed that both dffuson and jump-dffuson processes can be approxmated by weak GARCH models sampled at any dscrete tme frequency, wth the kurtoss of GARCH jump-dffuson processes beng larger than that of GARCH dffuson processes.

7 L. Grossmass and S.-H. Poon: Estmatng dynamc copula dependence usng ntraday data 57 tradng day becomes lmted (for example, f mn returns are used, there are approxmately 39 observaton ponts wthn the day) and the dependence estmate ncurs fnte samplng basness. To deal wth the multple sources of basness smultaneously, we consder the use of a rescaled ntraday dependence [smlar to that used n Martens, Chang, and Taylor () and Wggns (99) for volatlty] n our emprcal secton for VaR estmaton (see Secton 4.3). We denote the rescalng or adjustment factor by a f and estmate t as the average of the dependence ratos ˆ m θ / ˆ θ n the ntal n-sample perod. The adjusted ntraday estmator s the out-of-sample perods s then gven by ˆ m θ / a, where a f s assumed to be constant. f 3 Smulatons In ths secton, we study two cases to observe the effect of tme aggregaton on the ntraday dependence estmator proposed n Secton. In the frst case, returns are smulated under a SCOMDY model. In the second case we have a more neutral and realstc settng by usng a bvarate Normal Inverse Gaussan (NIG) Lévy process, where only the true lnear dependence s known. For our frst smulaton case, we generate random samples of sze 5, from the Gaussan, Clayton and survval Gumbel (s. Gumbel) copulas wth true copula parameters set at θ gaussan =.7. θ clayton =.5 and θ s.gumbel = 3.. To obtan the unvarate returns seres, we assume a zero condtonal mean and GARCH(,) parameters ω = ω = 3, α, = α, =.5 and β, = β, =.9, where the error terms are smulated from the respectve copulas above and normal margnals are assumed. The smulated returns are aggregated by k =,, 5,, 5,, 3, 6 and 9 tmes to form lower frequency returns samples. The GARCH and copula dependence parameters are then re-estmated at each aggregaton level. Table shows estmates of the mean and standard devaton of the pseudo maxmum lkelhood estmator for the Gaussan, Clayton and s. Gumbel. (The complete results wth the estmated GARCH parameters are gven n the Web Appendx Secton C.) As dscussed n Secton, the dependence parameter θ s relatvely stable for the Gaussan copula whle for fat-taled copulas, dependence declnes as the number of tmes of aggregaton ncreases. Ths s due to the decreasng kurtoss of the resduals as aggregaton tmes ncreases. For the second case, we repeat the smulaton usng a fat-taled bvarate NIG Lévy process, a sub-class of the more general class of hyperbolc Lévy processes. Ths class of processes are closed under convoluton unlke GARCH processes. The NIG Lévy process was ntroduced by Barndorff-Nelsen (995) as a realstc model for log stock prce returns. It exhbts non-gaussan sem-heavy tals and allows for skewness. We hence choose to smulate the NIG process as t s a contnuous model, t generates jumps whch results n fattaled dstrbutons, s closed under convoluton and has been found to be a realstc model for stock returns. Wth the excepton of the Gaussan case, the exact relatonshp between the copula dependence parameter and the NIG processes s unknown to us, but generally, a larger tal ndex and scale parameter n the process specfcaton would result n greater copula dependence. The bvarate process s gven by Table Estmaton results for copula dependence parameters under reestmaton at,, 5,, 5,, 3, 6 and 9 tmes of aggregaton. Copula True Aggregaton by k tmes Gaussan (.57) (.4) (.84) (.63) (.84) (.94) (.84) (.7) (.89) Clayton (.) (3.9) (.6) (.) (9.93) (.4) (.) (.8) (.7) s. Gumbel (5.43) (8.4) (7.7) (7.39) (8.9) (7.9) (7.35) (7.6) (7.5) Numbers n brackets are sample varance no. of sample observatons after aggregaton.

8 58 L. Grossmass and S.-H. Poon: Estmatng dynamc copula dependence usng ntraday data / X= µ + ZΓβ+ ZΓ Y, where β R, Z~IG[δ, α β T Γβ], IG[χ, ψ], α, δ, χ, ψ >. Γ R d d s a postve defnte structure matrx that controls the correlatons between the varables. IG denotes the nverse Gaussan dstrbuton whch has the pdf / f z χ e πz z z = + z χψ ( ) exp ( χ ψ ), >, 3 and Y~N d [,I]. The characterstc functon of the NIG process s gven by φ α β δ µ δ α β α β µ ( u;,,, ) = exp{ ( ( + u) ) + u}, where u R, µ R, δ>, β α, and the α, β, δ, μ parameters can be nterpreted as the tal ndex, symmetry, scale and locaton parameters, respectvely (Øgard et al. 5). Table gves the results of the estmated dependence under aggregaton of the smulated NIG process. 7 The Gaussan copula dependence s accurate at approxmately.5 throughout the aggregatons. Whle we do not have a true theoretcal value of θ for the survval Gumbel and Clayton copulas, we observe that there s a greater degree of stablty n the NIG smulatons as compared to the GARCH smulatons. There s stll a downward bas for survval Gumbel and a relatvely greater downward bas for Clayton copula dependence as the number of tmes of aggregaton ncreases. The two smulaton cases show that for copulas wth tal dependence, there s an upward bas n the estmaton of θ as samplng frequency ncreases (and samplng ntervals get smaller). Ths s n lne wth our dscusson n Secton.. Overall, the results are promsng n that despte the bas, the ntraday dependence estmator s nformatve about the true dependence of the data generatng process. Table Estmaton results for Gaussan, Clayton and s. Gumbel copula dependence when a heavy-taled bvarate NIG process s smulated. Copula True Aggregaton by k tmes Gaussan (37.6) (6.9) (.664) (.579) (.55) (.57) (.6) (.578) (.555) Clayton (43.5) (.3) (.5) (6.86) (5.67) (5.85) (4.43) (3.68) (3.65) s. Gumbel (3.8) (6.73) (.48) (.4) (.3) (.3) (.4) (.35) (.3) Numbers n brackets are sample varance no. of sample observatons after aggregaton. 4 Emprcal study In ths secton, we estmate VaR, an often-used rsk measure n fnancal nsttutons and provde emprcal evdence that the ntraday dependence estmator provdes more accurate out-of-sample VaR forecasts The followng parameters are used: α=, β, δ.89 = = smulated process has an emprcal mean =,. covarance = , kurtoss =..767 These values are close to those obtaned emprcally for our dataset..45 and µ =..43 The resultng skewness =.7, and excess.386

9 L. Grossmass and S.-H. Poon: Estmatng dynamc copula dependence usng ntraday data 59 than the constant, rollng wndow and autoregressve dependence estmators. We should emphasze that our focus s not on copula choce, but on comparng the performance of our tme-varyng dependence estmator wth other commonly-used tme-varyng dependence estmators. 4. Data For our emprcal study, we use the NYSE TAQ data of three stocks Ctgroup (c), JP Morgan (jpm) and IBM (bm) between January 6 and 3 December 8. The data s cleaned usng the algorthm descrbed n Barndorff-Nelsen et al. (8), and we nclude only observatons for each tradng day from 9.3 to 6.. We use quote data for our study as t contans a larger number of observatons as compared to trade data, thus reducng the effect of asynchronous tradng on the estmatons. Fnally, we use the calendar tme samplng and the last-tck nterpolaton method to obtan regularly spaced return observatons. Fgure plots the daly and -mn ntraday log returns tme seres of the three stocks. We can see that for both daly and ntradaly data, the volatlty of returns ncreases sharply from md-7 onwards due to the subprme crss. 4.. Intraday seasonalty adjustment and other ssues An mportant characterstc of hgh frequency data s the presence of ntraday seasonalty. Wood, McInsh, and Ord (985) and Harrs (986) were the frst to note ths perodc U-shape pattern n return volatlty durng the tradng day whle Andersen and Bollerslev (997) show that t s mportant to take nto account ntraday seasonalty before applyng standard volatlty models to ntraday data. They propose the Flexble Fourer Form (FFF) for estmatng the seasonalty pattern. Martens, Chang, and Taylor () show that FFF adjusted GARCH(,) forecasts ntraday volatlty better than other seasonalty correcton methods. We hence adopt FFF for seasonalty adjustment. To llustrate the resultng ntraday dependence estmate for both wth and wthout seasonalty adjustment, Fgure compares the BNS realzed correlaton 8 and the Gaussan copula ntraday dependence of Ctgroup and JP Morgan returns n the perod 6 8 usng -mn samplng frequency. The dotted lne s when the FFF seasonalty adjustment s appled whle the dash-dot lne s when no seasonalty adjustment s made. The graph shows the ntraday Gaussan dependence and the realzed correlaton to be closely matched. Ths means that dependence s preserved after the unvarate ARMA-GARCH fltraton, as was observed by Das and Embrechts (4). Furthermore, we fnd that the ncluson of seasonal adjustment gves a dependence parameter that s very close to that obtaned wthout seasonal adjustment. We also fnd that assumng a zero condtonal mean gves a dependence parameter that s close to that when the condtonal mean s modeled by an ARMA process (whch s used n Fgure ). We use FFF to adjust for ntraday seasonalty of the returns before passng them through a condtonal mean and volatlty model. 9 We fnd, however, that the seasonalty correcton removes the extreme observatons of each day (usually at the start and end of day), resultng n an ntraday resdual emprcal dstrbuton that has thnner tals. Fgure 3 shows the scatter plots of the ntraday resduals for the c-jpm stock par for three dfferent days: 3 January 6, August 7 and 3 October 8. The resdual plots on the left sde do not have deseasonalzaton adjustment whle those on the rght sde are when the returns are frst deseasonalzed. 8 The realzed covarance matrx that was used to compute realzed correlaton has been subsampled to reduce the effects of market mcrostructure nose. 9 For FFF as descrbed n Andersen and Bollerslev (997), we use P = and J = for c and jpm and P = 4 and J = for bm, and fnd that ths provdes an adequate ft. Fgure n the Web Appendx shows that the estmated average ntraday perodc patterns over the sample perod usng FFF provde a close approxmaton to the ntraday absolute returns.

10 5 L. Grossmass and S.-H. Poon: Estmatng dynamc copula dependence usng ntraday data.5 Ctgroup daly log returns Ctgroup mn log returns JP Morgan Chase daly log returns 6 8 IBM daly log returns JPM mn log returns (6 8) IBM mn log returns Fgure Return seres of c, jpm and bm (from top to bottom) at daly (left) and -mn (rght) frequences. A frst observaton s that the dependence of the stock par s clearly tme-varyng. On 3rd January 6, the scatter plots are relatvely ellptcal wth lttle tal dependence. On August 7, whch s close to the start of the credt crss, the dependence of the stock-par has ncreased sharply. 3 October 8, about the mddle of the credt crss, shows a reducton n lnear dependence as compared to August 7, but stll exhbts lower tal dependence. These dependence varatons over tme would have not been captured f we had used daly data though the perod (see Fgure 4).

11 L. Grossmass and S.-H. Poon: Estmatng dynamc copula dependence usng ntraday data 5.9 Realzed correlaton and ntraday dependence at mn samplng freq.8.7 C JPM dependence Intraday dep (Gaussan) Intraday dep w/ seasonal adj Realzed correlaton Fgure Plot of the Ctgroup-JP Morgan realzed correlaton (blue dotted lne) and the ntraday dependence of the Gaussan copula usng -mn samplng frequency wthout pre-applyng seasonalty adjustment (black sold lne wth *) and when FFF seasonalty adjustment s appled (red sold lne). The second observaton s that the shapes of the scatter plots are smlar when deseasonalzaton s appled (plots on rght) and when t s not (plots on left), hence resultng n the dependence parameter estmates for deseasonalzed and non-deseasonalzed returns beng rather close, as noted n Fgure. However, the densty of tal observaton s slghtly thnner after deseasonalzaton as t removes the effect of the begnnng and end of day returns, whch tend to exhbt the largest ntraday volatltes. Snce these extreme tal observatons are mportant for copula tal estmaton, we decde to not apply deseasonalzaton to the returns before passng through the ARMA-GARCH flter. Andersen and Bollerslev (997), Martens, Chang, and Taylor () and other papers apply the ARMA- GARCH fltraton to hgh frequency returns across multple days and fnd a large bas n the estmated model parameters nduced by the perodcty effect when seasonalty s not frst adjusted for. In ths paper, however, the condtonal mean and varance model s appled only to a sngle day s ntraday return and reestmated everyday. We fnd that ths provdes some form of adjustment for the ntraday seasonalty pattern. Fgure 5 shows the average ntraday condtonal varance of the stocks c, jpm and bm captured by GARCH(,). They exhbt the sgnature lopsded U-shape where volatlty s the hghest at the begnnng of the tradng day, lowest at mdday, and pcks up towards tradng day close. Hence standardsng returns usng a GARCH(,) s capable of removng the ntradaly seasonalty effect. An ssue that arses wth such an approach s that the GARCH varance looks non-statonary. However, we checked the estmated GARCH parameters and fnd them to fulfl the statonarty condtons. We next turn to the ssue of usng ARMA to model the condtonal mean. As wth many earler research papers, we fnd that the ncluson of the condtonal mean has a neglgble effect as compared to the condtonal varance. The magntude of the condtonal mean of ntraday data s very small but matter more than n the case of daly data. Andersen and Bollerslev (997) fnd that allowng for an MA term n the condtonal mean can account for the economcally mnor frst order autocorrelaton n returns, and the MA term s negatve at samplng frequences of less than 5 mn. The negatve MA term s sometmes explaned by the effect Smlar graphs for jpm and bm are found n the Web Appendx Secton A.

12 5 L. Grossmass and S.-H. Poon: Estmatng dynamc copula dependence usng ntraday data 4 Standardzed resduals on 3rd Jan 6 ( mn returns) 4 Standardzed resduals on 3rd Jan 6 (deseasonalzed mn returns) 3 3 JP Morgan - JP Morgan Ctgroup Ctgroup 4 Standardzed resduals on st Aug 7 ( mn returns) 4 Standardzed resduals on st Aug 7 (deseasonalzed mn returns) 3 3 JP Morgan JP Morgan Ctgroup Ctgroup 5 Standardzed resduals on 3 Oct 8 ( mn returns) 4 Standardzed resduals on 3 Oct 8 (deseasonalzed mn returns) JP Morgan - JP Morgan Ctgroup Ctgroup Fgure 3 Scatter plots of -mn resduals for c-jpm returns par for days 3 January 6 (top), August 7 (mddle) and 3 October 8 (bottom), respectvely. Plots of left sde use seasonalty-unadjusted returns whle plots on rght sde use deseasonalzed returns.

13 L. Grossmass and S.-H. Poon: Estmatng dynamc copula dependence usng ntraday data 53 4 Standardzed resduals (daly returns) 3 JP Morgan Ctgroup Fgure 4 Daly resdual scatter plot of stock par c-jpm for the perod between 3 January 6 and 3 December Average ntraday varance usng GARCH(,) C 7 6 Intraday varance Seconds after mdnght 4 Fgure 5 Average ntraday varance of c usng GARCH (,) effectvely captures the ntradaly seasonalty effects. Graphs for other stocks are gven n the Web Appendx. of bd-ask bounce n hgh frequency data due to dealers tradng around the spread producng a slght mean reverson n returns. We model the condtonal mean for ntraday returns usng an ARMA process and fnd the estmated dependence measures to be very close to the case where the condtonal mean s assumed to be zero. The resdual dstrbutons are smlar but slghtly more centred when the condtonal mean s fltered out. Snce our am s not to forecast ntraday returns, and the Ljung-Box statstcs of the resduals do not detect sgnfcant autocorrelatons after a GARCH fltraton, we do not flter out the condtonal mean for ntraday returns. However, for the hghly lqud stocks that we are usng, the effect of the bd-ask bounce s small, snce we are not consderng tck-by-tck returns. We also recognze that EGARCH or smlar models that account for leverage effects may be more approprate for daly equty data. We fnd, however, by plots of the squared resduals aganst the lagged resduals, that the leverage effect s not promnent n ntraday data.

14 54 L. Grossmass and S.-H. Poon: Estmatng dynamc copula dependence usng ntraday data 4.. Descrptve statstcs Descrptve statstcs of the return seres and GARCH resduals at daly and -mn samplng frequences are gven n Table 3. The descrptve statstcs ndcate that ntraday returns tend to have thnner tals than daly returns. However, the GARCH resduals dsplay the reverse characterstc [as found by Nelson (99) and Drost and Werker (996)]. Fnally, the Ljung-Box statstcs ndcate that there s no sgnfcant autocorrelaton n the resduals at both daly and ntradaly frequences Effect of samplng frequency To nvestgate the effects of samplng frequency on the ntraday dependence estmator when emprcal data s used, we estmate ntraday dependence of the stock pars c-jpm, c-bm and jpm-bm at samplng frequences rangng from to 3 mn. Fgure 6 shows the average ntraday dependence at the dfferent samplng frequences for year 6 (left) and year 8 (rght) for the stock-pars usng the Gaussan, survval Gumbel and Clayton copulas. We use these three copulas snce they are commonly used n emprcal applcatons. The graphs show that the parameters are farly stable at dfferent samplng frequences but show a consstent slght upward trend as the samplng frequency decreases. (Note that the Gaussan dependence parameter s also ncreasng slghtly but ths ncrease s less obvous n the graphs due to scale effect.) Whle Epps effects may be present at hgh samplng frequences, we postulate that ths upward bas s manly due to the Table 3 Descrptve statstcs of daly returns and unvarate GARCH resduals (top) and average descrptve statstcs at mn samplng frequency (bottom). c jpm bm Returns Resduals Returns Resduals Returns Resduals Daly frequency Mean Medan Std. Dev Skew Kurtoss Mn Max LB () (.) (.38) (.) (.4) (.) (.) LB (5) (.) (.69) (.) (.36) (.) (.3) Mn frequency Mean Medan Std. Dev Skew Kurtoss Mn Max LB () (.7) (.47) (.7) (.47) (.) (.4) LB (5) (.9) (.5) (.6) (.48) (.4) (.46) Numbers n brackets ndcate the p-value of the Ljung-Box statstc wth the null hypothess of no autocorrelaton. c, jpm and bm are short for Ctgroup, JP Morgan and IBM, respectvely.

15 L. Grossmass and S.-H. Poon: Estmatng dynamc copula dependence usng ntraday data 55 Average θ C vs. JPM (6) Average θ JPM vs. IBM (6) Average θ C vs. IBM (6) Gaussan Clayton Survval Gumbel Samplng frequency n mn Gaussan Clayton Survval Gumbel Samplng frequency n mn Gaussan Clayton Survval Gumbel Samplng frequency n mn Average θ C vs. JPM (8) Average θ JPM vs. IBM (8) Average θ C vs. IBM (8) Gaussan Clayton Survval Gumbel Samplng frequency n mn Gaussan Clayton Survval Gumbel Samplng frequency n mn Gaussan Clayton Survval Gumbel Samplng frequency n mn Fgure 6 Intraday dependence sgnature plots of c-jpm (top), c-bm (mddle) and jpm-bm (bottom) for year 6 and 8 usng the Gaussan (sold lnes), Clayton (dashed lnes) and s. Gumbel (x-lnes) copulas. effect of fnte samplng. 3 At -mn samplng frequency, there are 39 observatons for estmaton whle at 3 mn samplng frequency, there are only observatons for the estmaton of the dependence parameter. We now see that there are two potental effects that overlap when we use actual emprcal data: at too low samplng frequences we have fewer data ponts and face a postve bas due to fnte samplng, whle at very hgh samplng frequences, we face a postve bas due to the dstrbuton of the resduals tendng to fatter tals, as descrbed n Nelson (99). As a compromse of these two effects n the data, we use the -mn samplng frequency. Naturally an nterestng research queston would be the determnaton of an optmal samplng frequency but we do not delve further nto ths ssue here Copula choce Gven that there are hundreds of copula specfcatons, t s mpossble to consder all of them. To reduce the estmaton load, we estmate seven dfferent copulas [Gaussan, Clayton, survval Gumbel, Plackett, Frank, 3 See smulaton results n Secton C. n the Web Appendx.

16 56 L. Grossmass and S.-H. Poon: Estmatng dynamc copula dependence usng ntraday data Student s-t and symmetrzed Joe-Clayton (SJC)] usng the daly GARCH resduals of the three stock pars (c-jpm, c-bm, jpm-bm) and pck the three copulas that gve the three largest average log lkelhoods. The Gaussan, Plackett and Frank copulas do not exhbt tal dependence, whle the Clayton and survval Gumbel copulas have lower tal dependence. The Student s-t and SJC copulas have both upper and lower tal dependence. Table 4 gves the average log lkelhood estmates for the three stock pars. Fnally, we always nclude the Gaussan copula as a bass of comparson, whch gves four copulas for each stock par. Our treatment here on the ssue of copula choce s rather smplstc as t s not the focus of ths paper. Ths ssue s however mportant and non-trval, wth an ongong strand of research on copula selecton and goodness-of-ft testng (see, e.g., Dks, Panchenko, and van Djk () and Dks et al. (4) that optmzes copula choce usng out-of-sample forecastng crteron) and we refer the reader to these related lterature. 4. Comparng the ntraday dependence estmator wth tme-varyng dependence estmators We compare the ntraday dependence wth the constant uncondtonal dependence estmator and two other tme-varyng dependence estmators: the rollng wndow estmator and the autoregressve estmator. The constant uncondtonal dependence estmator estmates the copula parameter over the whole tme seres and one constant estmate over the whole perod s thus obtaned. The rollng wndow estmator estmates the copula parameter over a rollng wndow of 5 observatons (approxmately year). After the frst 5 observatons, the wndow ncorporates one new observaton and drops the oldest observaton at each ncremental tme pont. The autoregressve estmator s adapted from that of Patton (6) and estmated usng maxmum lkelhood. For the specfcaton of the Gaussan and SJC tme varyng copulas, we use the specfcaton of ARMA(,) as n Patton (6). As noted n the paper, a key dffculty s n dentfyng the forcng (or updatng) varable. For the other tme varyng copula models, we use the mean absolute dfference between the GARCH resduals u t and v t as forcng varable wth evoluton equaton for θ gven by θ = Λ β + β θ + β u v, (9) t t t j t j j= where Λ (x) = x + whch keeps the parameter above at all tmes. Ths s sutable for the Clayton, Gumbel, Student s- t s v parameter where the parameter must le above. 4 For the tme-varyng Plackett copula, the same forcng varable was used and Λ (x) = x. Snce for the Plackett copula, θ > and θ, penaltes to the lkelhood were mposed to deal wth these restrctons. Fgure 7 shows the estmated dependence measures of c-jpm usng dfferent copulas wth the constant estmator (sold horzontal lnes), ntraday dependence estmator (dotted lnes), rollng wndow estmator (dashed lnes) and autoregressve estmator (sold lnes). Overall, the graphs show that dependence between stocks s dynamc, and that the tme-varyng copula parameters tend to trend together. Ths emphaszes the phenomenon of tme-varyng dependence. In general, the autoregressve estmator hovers around the uncondtonal constant estmator whle the rollng wndow estmator dsplays a smooth trend. The ntraday Table 4 Average log lkelhood of dfferent copulas over ntraday dependence estmated for the perod 6 8. Gaussan Clayton s. Gumbel Plackett Frank Student s-t SJC c-jpm c-bm jpm-bm Bold font ndcates t s one of the best three log lkelhood estmates and s selected for further estmatons. 4 Clayton and survval Gumbel copulas do not allow for negatve dependence and have a parameter boundary value for θ of and, respectvely, whle the degrees of freedom for a Student s-t copula should le above f we assume fnte varance.

17 L. Grossmass and S.-H. Poon: Estmatng dynamc copula dependence usng ntraday data 57.9 Gaussan copula.8 Gumbel copula.8.6 Dependence of C JPM Dependence of C JPM Student s t copula rho and nu. sjc copula upper and lower tal nu for C JPM rho for C JPM Upper tal for C JPM Lower tal for C JPM Fgure 7 Constant and tme varyng dependence estmators Co (black sold lnes), RW (black dashed-dotted lnes), AR (blue sold lnes) and ID (red dotted lnes) for c-jpm between 6 and 8. Top row: Gaussan and s. Gumbel copulas. Left mddle and bottom rows: Student s-t copula (rho and v); rght mddle and bottom rows: SJC copula (upper and lower tal parameters). dependence estmates tends to be lower than other estmators durng the calm perods, but becomes greater than other estmators durng the subprme crss perod. The ntraday dependence estmator exhbts steep ncreases from the second quarter of 7 to about last quarter of 8 (concdng wth the subprme crss). A large drop n dependence s observed n November 8 when Ctgroup experenced a one week crash n ts stock prces. The rollng wndow estmator does effectvely capture the ncreasng dependence but exhbts a lag. The autoregressve estmator moves much n tandem wth the ntraday dependence measure n capturng short-term trends but on a long term bass moves largely about the constant uncondtonal estmator and and does not capture the ncreasng dependence over the perod. Fnally, the ntraday dependence estmator tends to le below the other estmators but ncreases dramatcally durng the crss perod. Ths llustrates how t effectvely captures the ncrease n rsk quckly durng perods when stocks are exhbtng large

18 58 L. Grossmass and S.-H. Poon: Estmatng dynamc copula dependence usng ntraday data negatve co-movements. Smlar observatons can be made for c-bm and jpm-bm stock pars (see Fgures and n the Appendx). 5 From Fgure 7, we can also see that the use of an asymmetrc copula, the SJC copula, allows us to dentfy that the lower tal dependence ncreases more steeply than that of the upper tal dependence durng the crss. There are more sharp peaks n the lower tal dependence durng the crss. A further comment s that the degrees of freedom parameter of the Student s-t copula seem to jump a lot between and above (see bottom left of Fgure 7). At above degrees of freedom, the Student s-t copula approaches that of a normal dstrbuton and we see that such occurrences are more common n 6 and the frst half of 7. Due to the large scale between and, the degree of freedom plots appears very jumpy, but llustrates varyng degree of fat-taledness that changes each day (realzed measures are n general rather volatle). 4.3 Usng ntraday dependence for VaR estmaton We now consder usng the ntraday dependence estmator for VaR estmaton. We use data from January 6 to 3 December 6 for estmaton and forecast the perod between January 7 to 3 December 8. The out-of-sample forecast perod s partcularly challengng gven that t conssts manly of the credt crss, whereas the ntal n-sample perod conssts only of the calm market perod. However we wsh to observe how the ntraday dependence performs n relaton to other tme varyng dependence measures n tmes of extreme market fluctuatons. Snce VaR s the next perod s forecasted loss wth a certan (extreme) probablty, ts estmaton usng copulas requres a forecast of the next day s dependence Forecastng ntraday dependence Fgure 8 shows the sample autocorrelatons (SACFs) of the ntraday dependences n a Gaussan, s. Gumbel and SJC copula for the c-jpm stock par. 6 It shows the slow decay of the dependence measures, a property also exhbted by realzed volatlty. The strong persstence s suggestve of long memory behavor of the ntraday dependence. Long memory processes can be modeled parametrcally by fractonally ntegrated processes Sample autocorrelatons Realzed dependences of c jpm stock par Gaussan Gumbel SJC Upper tal SJC Lower tal Fgure 8 Sample autocorrelatons of the ntraday dependences for the c-jpm stock par usng the Gaussan (sold lne), s. Gumbel (dash-dot lne) and SJC (lne wth dot) copulas. 5 For the Student s-t copula n c-bm stock par, convergence for the autoregressve estmator could not be obtaned for the ν (degrees of freedom) parameter. Ths llustrates a shortcomng of the autoregressve estmator n that such a parametrc form of the dynamcs s sutable for only certan copula dependences. 6 The Student s-t copula s not ncluded here as the Student s-t correlaton s smlar to that of the Gaussan copula dependence.